Theorem suplocexprlemlub 7665

Description: Lemma for suplocexpr 7666. The putative supremum is a least upper bound. (Contributed by Jim Kingdon, 14-Jan-2024.)

Hypotheses

Ref	Expression
suplocexpr.m	⊢ (𝜑 → ∃𝑥 𝑥 ∈ 𝐴)
suplocexpr.ub	⊢ (𝜑 → ∃𝑥 ∈ P ∀𝑦 ∈ 𝐴 𝑦<P 𝑥)
suplocexpr.loc	⊢ (𝜑 → ∀𝑥 ∈ P ∀𝑦 ∈ P (𝑥<P 𝑦 → (∃𝑧 ∈ 𝐴 𝑥<P 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧<P 𝑦)))
suplocexpr.b	⊢ 𝐵 = ⟨∪ (1st “ 𝐴), {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢}⟩

Assertion

Ref	Expression
suplocexprlemlub	⊢ (𝜑 → (𝑦<P 𝐵 → ∃𝑧 ∈ 𝐴 𝑦<P 𝑧))

Distinct variable groups:   𝑦,𝐴,𝑧   𝑥,𝐴,𝑦   𝑧,𝐵   𝜑,𝑦,𝑧   𝜑,𝑥

Allowed substitution hints:   𝜑(𝑤,𝑢)   𝐴(𝑤,𝑢)   𝐵(𝑥,𝑦,𝑤,𝑢)

Proof of Theorem suplocexprlemlub

Dummy variable 𝑠 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 109	. . . 4 ⊢ ((𝜑 ∧ 𝑦<P 𝐵) → 𝑦<P 𝐵)
2		ltrelpr 7446	. . . . . . . 8 ⊢ <P ⊆ (P × P)
3	2	brel 4656	. . . . . . 7 ⊢ (𝑦<P 𝐵 → (𝑦 ∈ P ∧ 𝐵 ∈ P))
4	3	simpld 111	. . . . . 6 ⊢ (𝑦<P 𝐵 → 𝑦 ∈ P)
5	4	adantl 275	. . . . 5 ⊢ ((𝜑 ∧ 𝑦<P 𝐵) → 𝑦 ∈ P)
6	3	simprd 113	. . . . . 6 ⊢ (𝑦<P 𝐵 → 𝐵 ∈ P)
7	6	adantl 275	. . . . 5 ⊢ ((𝜑 ∧ 𝑦<P 𝐵) → 𝐵 ∈ P)
8		ltdfpr 7447	. . . . 5 ⊢ ((𝑦 ∈ P ∧ 𝐵 ∈ P) → (𝑦<P 𝐵 ↔ ∃𝑠 ∈ Q (𝑠 ∈ (2nd ‘𝑦) ∧ 𝑠 ∈ (1st ‘𝐵))))
9	5, 7, 8	syl2anc 409	. . . 4 ⊢ ((𝜑 ∧ 𝑦<P 𝐵) → (𝑦<P 𝐵 ↔ ∃𝑠 ∈ Q (𝑠 ∈ (2nd ‘𝑦) ∧ 𝑠 ∈ (1st ‘𝐵))))
10	1, 9	mpbid 146	. . 3 ⊢ ((𝜑 ∧ 𝑦<P 𝐵) → ∃𝑠 ∈ Q (𝑠 ∈ (2nd ‘𝑦) ∧ 𝑠 ∈ (1st ‘𝐵)))
11		simprrr 530	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦<P 𝐵) ∧ (𝑠 ∈ Q ∧ (𝑠 ∈ (2nd ‘𝑦) ∧ 𝑠 ∈ (1st ‘𝐵)))) → 𝑠 ∈ (1st ‘𝐵))
12		suplocexpr.b	. . . . . . . . . 10 ⊢ 𝐵 = ⟨∪ (1st “ 𝐴), {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢}⟩
13	12	fveq2i 5489	. . . . . . . . 9 ⊢ (1st ‘𝐵) = (1st ‘⟨∪ (1st “ 𝐴), {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢}⟩)
14		npex 7414	. . . . . . . . . . . . 13 ⊢ P ∈ V
15	14	a1i 9	. . . . . . . . . . . 12 ⊢ (𝜑 → P ∈ V)
16		suplocexpr.m	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∃𝑥 𝑥 ∈ 𝐴)
17		suplocexpr.ub	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∃𝑥 ∈ P ∀𝑦 ∈ 𝐴 𝑦<P 𝑥)
18		suplocexpr.loc	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∀𝑥 ∈ P ∀𝑦 ∈ P (𝑥<P 𝑦 → (∃𝑧 ∈ 𝐴 𝑥<P 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧<P 𝑦)))
19	16, 17, 18	suplocexprlemss 7656	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ⊆ P)
20	15, 19	ssexd 4122	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ V)
21		fo1st 6125	. . . . . . . . . . . . 13 ⊢ 1st :V–onto→V
22		fofun 5411	. . . . . . . . . . . . 13 ⊢ (1st :V–onto→V → Fun 1st )
23	21, 22	ax-mp 5	. . . . . . . . . . . 12 ⊢ Fun 1st
24		funimaexg 5272	. . . . . . . . . . . 12 ⊢ ((Fun 1st ∧ 𝐴 ∈ V) → (1st “ 𝐴) ∈ V)
25	23, 24	mpan 421	. . . . . . . . . . 11 ⊢ (𝐴 ∈ V → (1st “ 𝐴) ∈ V)
26		uniexg 4417	. . . . . . . . . . 11 ⊢ ((1st “ 𝐴) ∈ V → ∪ (1st “ 𝐴) ∈ V)
27	20, 25, 26	3syl 17	. . . . . . . . . 10 ⊢ (𝜑 → ∪ (1st “ 𝐴) ∈ V)
28		nqex 7304	. . . . . . . . . . 11 ⊢ Q ∈ V
29	28	rabex 4126	. . . . . . . . . 10 ⊢ {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢} ∈ V
30		op1stg 6118	. . . . . . . . . 10 ⊢ ((∪ (1st “ 𝐴) ∈ V ∧ {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢} ∈ V) → (1st ‘⟨∪ (1st “ 𝐴), {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢}⟩) = ∪ (1st “ 𝐴))
31	27, 29, 30	sylancl 410	. . . . . . . . 9 ⊢ (𝜑 → (1st ‘⟨∪ (1st “ 𝐴), {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢}⟩) = ∪ (1st “ 𝐴))
32	13, 31	syl5eq 2211	. . . . . . . 8 ⊢ (𝜑 → (1st ‘𝐵) = ∪ (1st “ 𝐴))
33	32	eleq2d 2236	. . . . . . 7 ⊢ (𝜑 → (𝑠 ∈ (1st ‘𝐵) ↔ 𝑠 ∈ ∪ (1st “ 𝐴)))
34	33	ad2antrr 480	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦<P 𝐵) ∧ (𝑠 ∈ Q ∧ (𝑠 ∈ (2nd ‘𝑦) ∧ 𝑠 ∈ (1st ‘𝐵)))) → (𝑠 ∈ (1st ‘𝐵) ↔ 𝑠 ∈ ∪ (1st “ 𝐴)))
35	11, 34	mpbid 146	. . . . 5 ⊢ (((𝜑 ∧ 𝑦<P 𝐵) ∧ (𝑠 ∈ Q ∧ (𝑠 ∈ (2nd ‘𝑦) ∧ 𝑠 ∈ (1st ‘𝐵)))) → 𝑠 ∈ ∪ (1st “ 𝐴))
36		suplocexprlemell 7654	. . . . 5 ⊢ (𝑠 ∈ ∪ (1st “ 𝐴) ↔ ∃𝑧 ∈ 𝐴 𝑠 ∈ (1st ‘𝑧))
37	35, 36	sylib 121	. . . 4 ⊢ (((𝜑 ∧ 𝑦<P 𝐵) ∧ (𝑠 ∈ Q ∧ (𝑠 ∈ (2nd ‘𝑦) ∧ 𝑠 ∈ (1st ‘𝐵)))) → ∃𝑧 ∈ 𝐴 𝑠 ∈ (1st ‘𝑧))
38		simprl 521	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦<P 𝐵) ∧ (𝑠 ∈ Q ∧ (𝑠 ∈ (2nd ‘𝑦) ∧ 𝑠 ∈ (1st ‘𝐵)))) → 𝑠 ∈ Q)
39	38	ad2antrr 480	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑦<P 𝐵) ∧ (𝑠 ∈ Q ∧ (𝑠 ∈ (2nd ‘𝑦) ∧ 𝑠 ∈ (1st ‘𝐵)))) ∧ 𝑧 ∈ 𝐴) ∧ 𝑠 ∈ (1st ‘𝑧)) → 𝑠 ∈ Q)
40		simprrl 529	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦<P 𝐵) ∧ (𝑠 ∈ Q ∧ (𝑠 ∈ (2nd ‘𝑦) ∧ 𝑠 ∈ (1st ‘𝐵)))) → 𝑠 ∈ (2nd ‘𝑦))
41	40	ad2antrr 480	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑦<P 𝐵) ∧ (𝑠 ∈ Q ∧ (𝑠 ∈ (2nd ‘𝑦) ∧ 𝑠 ∈ (1st ‘𝐵)))) ∧ 𝑧 ∈ 𝐴) ∧ 𝑠 ∈ (1st ‘𝑧)) → 𝑠 ∈ (2nd ‘𝑦))
42		simpr 109	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑦<P 𝐵) ∧ (𝑠 ∈ Q ∧ (𝑠 ∈ (2nd ‘𝑦) ∧ 𝑠 ∈ (1st ‘𝐵)))) ∧ 𝑧 ∈ 𝐴) ∧ 𝑠 ∈ (1st ‘𝑧)) → 𝑠 ∈ (1st ‘𝑧))
43		rspe 2515	. . . . . . . 8 ⊢ ((𝑠 ∈ Q ∧ (𝑠 ∈ (2nd ‘𝑦) ∧ 𝑠 ∈ (1st ‘𝑧))) → ∃𝑠 ∈ Q (𝑠 ∈ (2nd ‘𝑦) ∧ 𝑠 ∈ (1st ‘𝑧)))
44	39, 41, 42, 43	syl12anc 1226	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑦<P 𝐵) ∧ (𝑠 ∈ Q ∧ (𝑠 ∈ (2nd ‘𝑦) ∧ 𝑠 ∈ (1st ‘𝐵)))) ∧ 𝑧 ∈ 𝐴) ∧ 𝑠 ∈ (1st ‘𝑧)) → ∃𝑠 ∈ Q (𝑠 ∈ (2nd ‘𝑦) ∧ 𝑠 ∈ (1st ‘𝑧)))
45	4	ad4antlr 487	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑦<P 𝐵) ∧ (𝑠 ∈ Q ∧ (𝑠 ∈ (2nd ‘𝑦) ∧ 𝑠 ∈ (1st ‘𝐵)))) ∧ 𝑧 ∈ 𝐴) ∧ 𝑠 ∈ (1st ‘𝑧)) → 𝑦 ∈ P)
46	19	ad4antr 486	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑦<P 𝐵) ∧ (𝑠 ∈ Q ∧ (𝑠 ∈ (2nd ‘𝑦) ∧ 𝑠 ∈ (1st ‘𝐵)))) ∧ 𝑧 ∈ 𝐴) ∧ 𝑠 ∈ (1st ‘𝑧)) → 𝐴 ⊆ P)
47		simplr 520	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑦<P 𝐵) ∧ (𝑠 ∈ Q ∧ (𝑠 ∈ (2nd ‘𝑦) ∧ 𝑠 ∈ (1st ‘𝐵)))) ∧ 𝑧 ∈ 𝐴) ∧ 𝑠 ∈ (1st ‘𝑧)) → 𝑧 ∈ 𝐴)
48	46, 47	sseldd 3143	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑦<P 𝐵) ∧ (𝑠 ∈ Q ∧ (𝑠 ∈ (2nd ‘𝑦) ∧ 𝑠 ∈ (1st ‘𝐵)))) ∧ 𝑧 ∈ 𝐴) ∧ 𝑠 ∈ (1st ‘𝑧)) → 𝑧 ∈ P)
49		ltdfpr 7447	. . . . . . . 8 ⊢ ((𝑦 ∈ P ∧ 𝑧 ∈ P) → (𝑦<P 𝑧 ↔ ∃𝑠 ∈ Q (𝑠 ∈ (2nd ‘𝑦) ∧ 𝑠 ∈ (1st ‘𝑧))))
50	45, 48, 49	syl2anc 409	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑦<P 𝐵) ∧ (𝑠 ∈ Q ∧ (𝑠 ∈ (2nd ‘𝑦) ∧ 𝑠 ∈ (1st ‘𝐵)))) ∧ 𝑧 ∈ 𝐴) ∧ 𝑠 ∈ (1st ‘𝑧)) → (𝑦<P 𝑧 ↔ ∃𝑠 ∈ Q (𝑠 ∈ (2nd ‘𝑦) ∧ 𝑠 ∈ (1st ‘𝑧))))
51	44, 50	mpbird 166	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑦<P 𝐵) ∧ (𝑠 ∈ Q ∧ (𝑠 ∈ (2nd ‘𝑦) ∧ 𝑠 ∈ (1st ‘𝐵)))) ∧ 𝑧 ∈ 𝐴) ∧ 𝑠 ∈ (1st ‘𝑧)) → 𝑦<P 𝑧)
52	51	ex 114	. . . . 5 ⊢ ((((𝜑 ∧ 𝑦<P 𝐵) ∧ (𝑠 ∈ Q ∧ (𝑠 ∈ (2nd ‘𝑦) ∧ 𝑠 ∈ (1st ‘𝐵)))) ∧ 𝑧 ∈ 𝐴) → (𝑠 ∈ (1st ‘𝑧) → 𝑦<P 𝑧))
53	52	reximdva 2568	. . . 4 ⊢ (((𝜑 ∧ 𝑦<P 𝐵) ∧ (𝑠 ∈ Q ∧ (𝑠 ∈ (2nd ‘𝑦) ∧ 𝑠 ∈ (1st ‘𝐵)))) → (∃𝑧 ∈ 𝐴 𝑠 ∈ (1st ‘𝑧) → ∃𝑧 ∈ 𝐴 𝑦<P 𝑧))
54	37, 53	mpd 13	. . 3 ⊢ (((𝜑 ∧ 𝑦<P 𝐵) ∧ (𝑠 ∈ Q ∧ (𝑠 ∈ (2nd ‘𝑦) ∧ 𝑠 ∈ (1st ‘𝐵)))) → ∃𝑧 ∈ 𝐴 𝑦<P 𝑧)
55	10, 54	rexlimddv 2588	. 2 ⊢ ((𝜑 ∧ 𝑦<P 𝐵) → ∃𝑧 ∈ 𝐴 𝑦<P 𝑧)
56	55	ex 114	1 ⊢ (𝜑 → (𝑦<P 𝐵 → ∃𝑧 ∈ 𝐴 𝑦<P 𝑧))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 698   = wceq 1343  ∃wex 1480   ∈ wcel 2136  ∀wral 2444  ∃wrex 2445  {crab 2448  Vcvv 2726   ⊆ wss 3116  ⟨cop 3579  ∪ cuni 3789  ∩ cint 3824   class class class wbr 3982   “ cima 4607  Fun wfun 5182  –onto→wfo 5186  ‘cfv 5188  1^st c1st 6106  2^nd c2nd 6107  Qcnq 7221   <_Q cltq 7226  Pcnp 7232  <_P cltp 7236

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-iinf 4565

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-1st 6108  df-qs 6507  df-ni 7245  df-nqqs 7289  df-inp 7407  df-iltp 7411

This theorem is referenced by:  suplocexpr  7666